# Statistical Inference for a General Family of Modified Exponentiated Distributions

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## Abstract

**:**

## 1. Introduction

- A variety of shapes can be obtained due to the introduction of parameter $\beta >0\phantom{\rule{0.277778em}{0ex}}$.
- The random variable (rv) X with cdf F given in (1) can be obtained by applying a kind of inverse probability integral transformation (or alternatively ${G}^{-1}$) to a $Beta(1/\beta ,1)$ distribution.

## 2. Materials and Methods

**Definition**

**1.**

#### Genesis of the Family

## 3. Results

#### 3.1. Quantile Function and Random Number Generation

**Proposition**

**1.**

**Proof.**

- Generate $u\sim \mathrm{Uniform}(0,1)$.
- Compute $y={Q}_{G}\left(\right)open="("\; close=")">{\left(\right)}^{\left(\right)}u+1-1;\eta $.

#### 3.2. Moments

**Proposition**

**2.**

**Proof.**

#### 3.3. Stochastic Interpretation of the Parameters

**Definition**

**2.**

**Proposition**

**3.**

**Proof.**

**Definition**

**3.**

- 1.
- ${X}_{1}$ is said to be smaller than ${X}_{2}$ in the hazard rate order, denoted by ${X}_{1}{\le}_{HR}{X}_{2},$ if ${h}_{1}\left(x\right)\ge {h}_{2}\left(x\right)$ for all x.
- 2.
- ${X}_{1}$ is said to be stochastically smaller than ${X}_{2}$, denoted by ${X}_{1}{\le}_{ST}{X}_{2},$ if ${F}_{1}\left(x\right)\ge {F}_{2}\left(x\right)$ for all x.

**Corollary**

**1.**

- 1.
- $X{\le}_{HR}Y$ and $X{\le}_{ST}Y$.
- 2.
- $E\left({X}^{k}\right)\le E\left({Y}^{k}\right)\phantom{\rule{0.277778em}{0ex}}$$\phantom{\rule{0.277778em}{0ex}}\forall k\in {\mathbb{Z}}^{+}$.

**Proof.**

#### 3.4. Poisson-Modified Exponentiated Mixture

**Proposition**

**4.**

**Proof.**

**Corollary**

**2.**

- 1.
- ${X}_{1}{\le}_{HR}{X}_{2}$ and ${X}_{1}{\le}_{ST}{X}_{2}$.
- 2.
- $E\left({X}_{1}^{k}\right)\le E\left({X}_{2}^{k}\right)$, for $k\in {\mathbb{Z}}^{+}$.

#### 3.5. Order Statistics

**Corollary**

**3.**

- 1.
- ${X}_{r:n}{\le}_{ST}{Y}_{r:n}\phantom{\rule{0.277778em}{0ex}}$ for $\phantom{\rule{0.277778em}{0ex}}r=1,\dots ,n$.
- 2.
- $E\left({X}_{r:n}\right)\le E\left({Y}_{r:n}\right)\phantom{\rule{0.277778em}{0ex}}$, provided the expectations exist.

**Proof.**

#### 3.6. Right Tail Behavior

**Lemma**

**1.**

**Proposition**

**5.**

**Proof.**

**Corollary**

**4.**

**Corollary**

**5.**

- 1.
- Take $G\left(y\right)={(\theta /y)}^{\lambda}$, $\phantom{\rule{0.277778em}{0ex}}y\ge \theta $, the cdf of a Pareto distribution with shape parameter $\lambda >0$ and scale parameter $\theta >0$. Then the $\mathrm{ME}\phantom{\rule{0.277778em}{0ex}}\mathrm{G}(\theta ,\lambda ,\beta )$ model is heavy-right-tailed.
- 2.
- Take $G\left(y\right)={\mathrm{\Phi}}_{\theta ,\lambda}(logy)$, $\phantom{\rule{0.277778em}{0ex}}y>0$; the cdf of a lognormal distribution with parameters $\theta \in \mathbb{R}$ and $\lambda >0$; and Φ to denote the cdf of the standard normal distribution. Then the $\mathrm{ME}\phantom{\rule{0.277778em}{0ex}}\mathrm{G}(\theta ,\lambda ,\beta )$ model is heavy-right-tailed.

**Proof.**

- In this case $g\left(y\right)=\lambda {\theta}^{\lambda}{y}^{-1-\lambda}$ and ${g}^{\prime}\left(y\right)/g\left(y\right)=-(1+\lambda )/y$. Both functions tend to zero as $y\to \infty $, $\phantom{\rule{0.277778em}{0ex}}\forall \lambda >0$.
- In this case $g\left(y\right)=1/\left(\lambda y\sqrt{2\pi}\right)exp\left(\right)open="["\; close="]">-{(logy-\theta )}^{2}/\left(2{\lambda}^{2}\right)$ and ${g}^{\prime}\left(y\right)/g\left(y\right)=(\theta -{\lambda}^{2}-logy)/\left(y{\lambda}^{2}\right)$, which tend to zero as $y\to \infty $.

**Definition**

**4.**

**Proposition**

**6.**

**Proof.**

**Corollary**

**6.**

**Corollary**

**7.**

**Proof.**

## 4. Inference

#### 4.1. Moment Estimators

#### 4.2. Maximum Likelihood Estimation

`optim`function available in the R package can be used in the maximization process. As initial values to start the process of estimation, the moment estimates can be used. Another option is to start with the MLEs of parameters in the parent distribution [28], ${\widehat{\eta}}_{0}$, and to propose an initial ${\widehat{\beta}}_{0}$ from one of the equations in (17).

## 5. A Relevant Sub-Model: The Exponential Case

**Definition**

**5.**

**Proposition**

**7.**

- 1.
- The hazard function is$$h(y;\alpha ,\beta )=\frac{\alpha \beta {e}^{-\alpha y}{\left(\right)}^{2}\beta -1}{}{2}^{\beta}-{\left(\right)}^{2}\beta ,\phantom{\rule{1.em}{0ex}}y0,\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\alpha 0,\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\beta 0\phantom{\rule{0.277778em}{0ex}}.$$
- 2.
- The moment generating function of Y is$$\begin{array}{c}\hfill {M}_{Y}\left(t\right)={2}^{\beta -t/\alpha}\phi \left(\beta \right){B}_{1/2}(1-t/\alpha ,\beta ),\phantom{\rule{2.em}{0ex}}t<\alpha ,\end{array}$$
- 3.
- The expected value of Y can be obtained as$$\begin{array}{c}\hfill E\left(Y\right)=\frac{{2}^{\beta -1}\phi \left(\beta \right)}{\alpha}{\phantom{\rule{0.166667em}{0ex}}}_{3}{F}_{2}(\{1,1,1-\beta \};\{2,2\};1/2),\end{array}$$$$\begin{array}{c}\hfill {}_{p}{F}_{q}(\{{a}_{1},\cdots ,{a}_{p}\};\{{b}_{1},\cdots ,{b}_{q}\};z)=\sum _{k=0}^{\infty}\frac{{\left({a}_{1}\right)}_{k}\cdots {\left({a}_{p}\right)}_{k}}{{\left({b}_{1}\right)}_{k}\cdots {\left({b}_{q}\right)}_{k}}\frac{{z}^{k}}{k!}\end{array}$$is the generalized hypergeometric function.

**Proof.**

**Remark**

**1.**

**Proposition**

**8**

- 1.
- For any $\alpha >0$, it is verified that if $\beta \le 2$, then $f(y;\alpha ,\beta )$ is monotically decreasing and its mode is at zero. On the other hand, if $\beta >2$, then $f(y;\alpha ,\beta )$ is strongly unimodal and the mode is at $\frac{1}{\alpha}\phantom{\rule{0.277778em}{0ex}}log\left(\right)open="("\; close=")">\frac{\beta}{2}$, where $log(\xb7)$ denotes natural logarithm.
- 2.
- For any $\alpha >0$, it is verified that if $\beta <1$, then the hrf, $h(y;\alpha ,\beta )$, is monotically decreasing; if $\beta =1$, then the hrf is constant, $h(y;\alpha ,\beta )=\alpha \phantom{\rule{0.277778em}{0ex}}$$\phantom{\rule{0.277778em}{0ex}}\forall y>0$; and if $\beta >1$, then $h(y;\alpha ,\beta )$ is monotically increasing.Moreover,$$\underset{y\to \infty}{lim}h(y;\alpha ,\beta )=\alpha \phantom{\rule{0.277778em}{0ex}}.$$

**Proof.**

**Proposition**

**9.**

**Proof.**

**Corollary**

**8.**

- 1.
- If $\beta >1$ ($\beta <1$), then the cdf and survival functions are log-concave (log-convex).
- 2.
- If $\beta >1$, then the truncated distribution at $c\in support\left(Y\right)$, ${f}^{c}\left(y\right)=f\left(y\right)/\overline{F}\left(y\right){1}_{\{y\ge c\}}$, is also log-concave.

**Proof.**

**Remark**

**2**

**Corollary**

**9.**

**Remark**

**3**

- 1.
- In reliability and survival analysis, models such as the $\mathrm{MEE}(\alpha ,\beta )$ distribution are of interest. In this context, quantiles are used to establish warranty periods of products, for modeling lifetime data, and to estimate features in the model not affected by the presence of outliers. The parametric estimation of quantiles tends to be more efficient than nonparametric estimation, as can be seen in [30].
- 2.
- In financial and actuarial theory, the quantile function is also known as the value at risk, (denoted as VaR), which is interpreted as the amount of capital required to ensure that the insurer (or the economic agent) does not become insolvent with a high degree of certainty. Therefore, it is of interest to have an explicit expression for this function, such as the one given in (26). This measure is also of great importance in scenarios where outliers corresponding to large empirical data may appear, which is quite common in risk theory. In this sense, we highlight that the MEE distribution seems appropriate for modeling this kind of empirical data, as can be seen in Figure 3.

**Corollary**

**10.**

## 6. Simulation Study

- The ML estimators were biased, but the bias tended to zero when n increased.
- The standard deviation of the estimates decreased when the sample size n increased.
- Since the previous summaries tended to zero if n increased, both estimators seem to have been consistent. That is, if $n\to \infty $, then closer estimates to the true values of the parameters were obtained.
- It can be appreciated that, for the values of the parameters and the sample sizes under consideration, the bias and standard deviation of $\widehat{\alpha}$ are greater than the bias and standard deviation of $\widehat{\beta}$, but in any case, both estimators exhibited good behavior, since if n increased, then both summaries decreased.

## 7. Application

#### Modeling of a Fatigue Fracture in Kevlar 373/Epoxy Data

- 1.
- Weibull, $W(\beta ,\lambda )$, distribution:$$f(x;\lambda ,\beta )=\frac{\beta}{{\lambda}^{\beta}}{x}^{\beta -1}{e}^{-{\left(\right)}^{\frac{x}{\lambda}}\beta}$$
- 2.
- Generalized exponential, $GE(\beta ,\lambda )$, distribution:$$f(x;\lambda ,\beta )=\beta \lambda {e}^{-\lambda x}{\left(\right)}^{1}\beta -1$$

## 8. Discussion

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

cdf | cumulative distribution function |

rv | random variable |

probability density function | |

hrf | hazard rate function |

MEG | modified exponentiated family with parent cdf G |

MEE | modified exponentiated exponential model |

W | Weibull |

GE | generalized exponential |

E | (classical) Exponential distribution |

ML | maximum likelihood |

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MEE | |||||||
---|---|---|---|---|---|---|---|

$\mathit{n}=\mathbf{50}$ | $\mathit{n}=\mathbf{100}$ | $\mathit{n}=\mathbf{200}$ | |||||

$\mathbf{\alpha}$ | $\mathbf{\beta}$ | $\widehat{\mathbf{\alpha}}$(s.e.) | $\widehat{\mathbf{\beta}}$(s.e.) | $\widehat{\mathbf{\alpha}}$(s.e.) | $\widehat{\mathbf{\beta}}$(s.e.) | $\widehat{\mathbf{\alpha}}$(s.e.) | $\widehat{\mathbf{\beta}}$(s.e.) |

3 | 2 | 3.150 (1.120) | 2.038 (0.363) | 3.084 (0.845) | 2.036 (0.257) | 3.070 (0.627) | 2.022 (0.191) |

4 | 4.154 (1.205) | 2.046 (0.325) | 4.117 (0.895) | 2.027 (0.245) | 4.050 (0.650) | 2.013 (0.163) | |

5 | 5.204 (1.230) | 2.043 (0.296) | 5.152 (0.963) | 2.026 (0.206) | 5.056 (0.696) | 2.010 (0.155) | |

10 | 5 | 10.743 (2.697) | 5.123 (0.668) | 10.426 (1.851) | 5.087 (0.455) | 10.163 (1.214) | 5.023 (0.316) |

10 | 10.627 (2.589) | 10.207 (1.248) | 10.346 (1.751) | 10.109 (0.864) | 10.136 (1.133) | 10.058 (0.624) | |

15 | 10.674 (2.449) | 15.350 (1.851) | 10.366 (1.824) | 15.225 (1.368) | 10.270 (1.218) | 15.146 (0.933) | |

3 | 3 | 3.255 (1.305) | 3.123 (0.580) | 3.137 (0.895) | 3.071 (0.398) | 3.071 (0.604) | 3.023 (0.271) |

4 | 4 | 4.241 (1.366) | 4.129 (0.695) | 4.107 (0.942) | 4.043 (0.490) | 4.065 (0.640) | 4.041 (0.342) |

5 | 5 | 5.291 (1.538) | 5.132 (0.791) | 5.138 (0.972) | 5.085 (0.544) | 5.019 (0.707) | 5.005 (0.389) |

0.0251 | 0.0886 | 0.0891 | 0.2501 | 0.3113 | 0.3451 | 0.4763 | 0.5650 |

0.5671 | 0.6566 | 0.6748 | 0.6751 | 0.6753 | 0.7696 | 0.8375 | 0.8391 |

0.8425 | 0.8645 | 0.8851 | 0.9113 | 0.9120 | 0.9836 | 1.0483 | 1.0596 |

1.0773 | 1.1733 | 1.2570 | 1.2766 | 1.2985 | 1.3211 | 1.3503 | 1.3551 |

1.4595 | 1.4880 | 1.5728 | 1.5733 | 1.7083 | 1.7263 | 1.7460 | 1.7630 |

1.7746 | 1.8275 | 1.8375 | 1.8503 | 1.8808 | 1.8878 | 1.8881 | 1.9316 |

1.9558 | 2.0048 | 2.0408 | 2.0903 | 2.1093 | 2.1330 | 2.2100 | 2.2460 |

2.2878 | 2.3203 | 2.3470 | 2.3513 | 2.4951 | 2.5260 | 2.9911 | 3.0256 |

3.2678 | 3.4045 | 3.4846 | 3.7433 | 3.7455 | 3.9143 | 4.8073 | 5.4005 |

5.4435 | 5.5295 | 6.5541 | 9.0960 |

n | $\overline{\mathit{y}}$ | s | ${\mathit{b}}_{1}$ | ${\mathit{b}}_{2}$ |
---|---|---|---|---|

76 | 1.959 | 1.574 | 1.979 | 8.161 |

Parameters Estimated | MEE (s.e.) | GE (s.e.) | W (s.e.) | E (s.e.) |
---|---|---|---|---|

$\widehat{\beta}$ | 4.807 (1.120) | 1.709 (0.282) | 1.325 (0.113) | - |

$\widehat{\lambda}$ | 0.831 (0.106) | 0.702 (0.092) | 2.132 (0.194) | 0.510 (0.058) |

Criterion | MEE | GE | W | E |
---|---|---|---|---|

AIC | 246.3844 | 248.4872 | 249.0494 | 256.2286 |

BIC | 251.0459 | 253.1487 | 253.7109 | 258.5593 |

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## Share and Cite

**MDPI and ACS Style**

Gómez-Déniz, E.; Iriarte, Y.A.; Gómez, Y.M.; Barranco-Chamorro, I.; Gómez, H.W.
Statistical Inference for a General Family of Modified Exponentiated Distributions. *Mathematics* **2021**, *9*, 3069.
https://doi.org/10.3390/math9233069

**AMA Style**

Gómez-Déniz E, Iriarte YA, Gómez YM, Barranco-Chamorro I, Gómez HW.
Statistical Inference for a General Family of Modified Exponentiated Distributions. *Mathematics*. 2021; 9(23):3069.
https://doi.org/10.3390/math9233069

**Chicago/Turabian Style**

Gómez-Déniz, Emilio, Yuri A. Iriarte, Yolanda M. Gómez, Inmaculada Barranco-Chamorro, and Héctor W. Gómez.
2021. "Statistical Inference for a General Family of Modified Exponentiated Distributions" *Mathematics* 9, no. 23: 3069.
https://doi.org/10.3390/math9233069